Theory Nat

(*  Title:      ZF/Nat.thy
    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
    Copyright   1994  University of Cambridge

section‹The Natural numbers As a Least Fixed Point›

theory Nat imports OrdQuant Bool begin

  nat :: i  where
    "nat  lfp(Inf, λX. {0}  {succ(i). i  X})"

  quasinat :: "i  o"  where
    "quasinat(n)  n=0 | (m. n = succ(m))"

  (*Has an unconditional succ case, which is used in "recursor" below.*)
  nat_case :: "[i, ii, i]i"  where
    "nat_case(a,b,k)  THE y. k=0  y=a | (x. k=succ(x)  y=b(x))"

  nat_rec :: "[i, i, [i,i]i]i"  where
          wfrec(Memrel(nat), k, λn f. nat_case(a, λm. b(m, f`m), n))"

  (*Internalized relations on the naturals*)

  Le :: i  where
    "Le  {x,y:nat*nat. x  y}"

  Lt :: i  where
    "Lt  {x, y:nat*nat. x < y}"

  Ge :: i  where
    "Ge  {x,y:nat*nat. y  x}"

  Gt :: i  where
    "Gt  {x,y:nat*nat. y < x}"

  greater_than :: "ii"  where
    "greater_than(n)  {i  nat. n < i}"

text‹No need for a less-than operator: a natural number is its list of

lemma nat_bnd_mono: "bnd_mono(Inf, λX. {0}  {succ(i). i  X})"
apply (rule bnd_monoI)
apply (cut_tac infinity, blast, blast)

(* @{term"nat = {0} ∪ {succ(x). x ∈ nat}"} *)
lemmas nat_unfold = nat_bnd_mono [THEN nat_def [THEN def_lfp_unfold]]

(** Type checking of 0 and successor **)

lemma nat_0I [iff,TC]: "0  nat"
apply (subst nat_unfold)
apply (rule singletonI [THEN UnI1])

lemma nat_succI [intro!,TC]: "n  nat  succ(n)  nat"
apply (subst nat_unfold)
apply (erule RepFunI [THEN UnI2])

lemma nat_1I [iff,TC]: "1  nat"
by (rule nat_0I [THEN nat_succI])

lemma nat_2I [iff,TC]: "2  nat"
by (rule nat_1I [THEN nat_succI])

lemma bool_subset_nat: "bool  nat"
by (blast elim!: boolE)

lemmas bool_into_nat = bool_subset_nat [THEN subsetD]

subsection‹Injectivity Properties and Induction›

(*Mathematical induction*)
lemma nat_induct [case_names 0 succ, induct set: nat]:
    "n  nat;  P(0);  x. x  nat;  P(x)  P(succ(x))  P(n)"
by (erule def_induct [OF nat_def nat_bnd_mono], blast)

lemma natE:
 assumes "n  nat"
 obtains ("0") "n=0" | (succ) x where "x  nat" "n=succ(x)"
using assms
by (rule nat_unfold [THEN equalityD1, THEN subsetD, THEN UnE]) auto

lemma nat_into_Ord [simp]: "n  nat  Ord(n)"
by (erule nat_induct, auto)

(* @{term"i ∈ nat ⟹ 0 ≤ i"}; same thing as @{term"0<succ(i)"}  *)
lemmas nat_0_le = nat_into_Ord [THEN Ord_0_le]

(* @{term"i ∈ nat ⟹ i ≤ i"}; same thing as @{term"i<succ(i)"}  *)
lemmas nat_le_refl = nat_into_Ord [THEN le_refl]

lemma Ord_nat [iff]: "Ord(nat)"
apply (rule OrdI)
apply (erule_tac [2] nat_into_Ord [THEN Ord_is_Transset])
  unfolding Transset_def
apply (rule ballI)
apply (erule nat_induct, auto)

lemma Limit_nat [iff]: "Limit(nat)"
  unfolding Limit_def
apply (safe intro!: ltI Ord_nat)
apply (erule ltD)

lemma naturals_not_limit: "a  nat  ¬ Limit(a)"
by (induct a rule: nat_induct, auto)

lemma succ_natD: "succ(i): nat  i  nat"
by (rule Ord_trans [OF succI1], auto)

lemma nat_succ_iff [iff]: "succ(n): nat  n  nat"
by (blast dest!: succ_natD)

lemma nat_le_Limit: "Limit(i)  nat  i"
apply (rule subset_imp_le)
apply (simp_all add: Limit_is_Ord)
apply (rule subsetI)
apply (erule nat_induct)
 apply (erule Limit_has_0 [THEN ltD])
apply (blast intro: Limit_has_succ [THEN ltD] ltI Limit_is_Ord)

(* ⟦succ(i): k;  k ∈ nat⟧ ⟹ i ∈ k *)
lemmas succ_in_naturalD = Ord_trans [OF succI1 _ nat_into_Ord]

lemma lt_nat_in_nat: "m<n;  n  nat  m  nat"
apply (erule ltE)
apply (erule Ord_trans, assumption, simp)

lemma le_in_nat: "m  n; n  nat  m  nat"
by (blast dest!: lt_nat_in_nat)

subsection‹Variations on Mathematical Induction›

(*complete induction*)

lemmas complete_induct = Ord_induct [OF _ Ord_nat, case_names less, consumes 1]

lemma complete_induct_rule [case_names less, consumes 1]:
  "i  nat  (x. x  nat  (y. y  x  P(y))  P(x))  P(i)"
  using complete_induct [of i P] by simp

(*Induction starting from m rather than 0*)
lemma nat_induct_from:
  assumes "m  n" "m  nat" "n  nat"
    and "P(m)"
    and "x. x  nat;  m  x;  P(x)  P(succ(x))"
  shows "P(n)"
proof -
  from assms(3) have "m  n  P(m)  P(n)"
    by (rule nat_induct) (use assms(5) in simp_all add: distrib_simps le_succ_iff)
  with assms(1,2,4) show ?thesis by blast

(*Induction suitable for subtraction and less-than*)
lemma diff_induct [case_names 0 0_succ succ_succ, consumes 2]:
    "m  nat;  n  nat;
        x. x  nat  P(x,0);
        y. y  nat  P(0,succ(y));
        x y. x  nat;  y  nat;  P(x,y)  P(succ(x),succ(y))
apply (erule_tac x = m in rev_bspec)
apply (erule nat_induct, simp)
apply (rule ballI)
apply (rename_tac i j)
apply (erule_tac n=j in nat_induct, auto)

(** Induction principle analogous to trancl_induct **)

lemma succ_lt_induct_lemma [rule_format]:
     "m  nat  P(m,succ(m))  (xnat. P(m,x)  P(m,succ(x))) 
                 (nnat. m<n  P(m,n))"
apply (erule nat_induct)
 apply (intro impI, rule nat_induct [THEN ballI])
   prefer 4 apply (intro impI, rule nat_induct [THEN ballI])
apply (auto simp add: le_iff)

lemma succ_lt_induct:
    "m<n;  n  nat;
        x. x  nat;  P(m,x)  P(m,succ(x))
by (blast intro: succ_lt_induct_lemma lt_nat_in_nat)

subsection‹quasinat: to allow a case-split rule for termnat_case

text‹True if the argument is zero or any successor›
lemma [iff]: "quasinat(0)"
by (simp add: quasinat_def)

lemma [iff]: "quasinat(succ(x))"
by (simp add: quasinat_def)

lemma nat_imp_quasinat: "n  nat  quasinat(n)"
by (erule natE, simp_all)

lemma non_nat_case: "¬ quasinat(x)  nat_case(a,b,x) = 0"
by (simp add: quasinat_def nat_case_def)

lemma nat_cases_disj: "k=0 | (y. k = succ(y)) | ¬ quasinat(k)"
apply (case_tac "k=0", simp)
apply (case_tac "m. k = succ(m)")
apply (simp_all add: quasinat_def)

lemma nat_cases:
     "k=0  P;  y. k = succ(y)  P; ¬ quasinat(k)  P  P"
by (insert nat_cases_disj [of k], blast)

(** nat_case **)

lemma nat_case_0 [simp]: "nat_case(a,b,0) = a"
by (simp add: nat_case_def)

lemma nat_case_succ [simp]: "nat_case(a,b,succ(n)) = b(n)"
by (simp add: nat_case_def)

lemma nat_case_type [TC]:
    "n  nat;  a  C(0);  m. m  nat  b(m): C(succ(m))
      nat_case(a,b,n)  C(n)"
by (erule nat_induct, auto)

lemma split_nat_case:
   ((k=0  P(a))  (x. k=succ(x)  P(b(x)))  (¬ quasinat(k)  P(0)))"
apply (rule nat_cases [of k])
apply (auto simp add: non_nat_case)

subsection‹Recursion on the Natural Numbers›

(** nat_rec is used to define eclose and transrec, then becomes obsolete.
    The operator rec, from arith.thy, has fewer typing conditions **)

lemma nat_rec_0: "nat_rec(0,a,b) = a"
apply (rule nat_rec_def [THEN def_wfrec, THEN trans])
 apply (rule wf_Memrel)
apply (rule nat_case_0)

lemma nat_rec_succ: "m  nat  nat_rec(succ(m),a,b) = b(m, nat_rec(m,a,b))"
apply (rule nat_rec_def [THEN def_wfrec, THEN trans])
 apply (rule wf_Memrel)
apply (simp add: vimage_singleton_iff)

(** The union of two natural numbers is a natural number -- their maximum **)

lemma Un_nat_type [TC]: "i  nat; j  nat  i  j  nat"
apply (rule Un_least_lt [THEN ltD])
apply (simp_all add: lt_def)

lemma Int_nat_type [TC]: "i  nat; j  nat  i  j  nat"
apply (rule Int_greatest_lt [THEN ltD])
apply (simp_all add: lt_def)

(*needed to simplify unions over nat*)
lemma nat_nonempty [simp]: "nat  0"
by blast

text‹A natural number is the set of its predecessors›
lemma nat_eq_Collect_lt: "i  nat  {jnat. j<i} = i"
apply (rule equalityI)
apply (blast dest: ltD)
apply (auto simp add: Ord_mem_iff_lt)
apply (blast intro: lt_trans)

lemma Le_iff [iff]: "x,y  Le  x  y  x  nat  y  nat"
by (force simp add: Le_def)